arXiv subjects

Rishi Saket

Publications and source records attributed to Rishi Saket.

At least 19 recordsLinked to original sources

arXiv2026

Label Differential Privacy via Aggregation

This paper explores the use of linear aggregation to protect the privacy of sensitive training labels through the concept of \emph{label differential privacy} (label-DP) while maintaining regression task utility. Our key finding is that weighted linear aggregation of training instances with i.i.d. $N(0, 1)$ weights can achieve $(\varepsilon, δ)$-label-DP with $m = O\left(n/(\log(1/δ))\right)$. Unlike prior methods, our approach relies on the minimum linear regression loss rather than the minimum singular value of the data matrix, resulting in better practical bounds on real datasets. We also examine real-world mechanisms involving disjoint sets or \textit{bags} of instances. We demonstrate that aggregating labels from sub-sampled disjoint $k$-sized bags using i.i.d. $N(0,1)$ weights achieves $(\varepsilon,δ)$-label-DP with $k \geq Ω\left(\left((1/\varepsilon)\log\left(1/δ\right)\right)^2\right)$. In both scenarios, the optimal linear mse-regressor on the aggregated data approximates the original dataset's optimum with high probability, without needing additive label noise. Furthermore, we show that adding $N(0,1)$ noise to any constant fraction of labels allows for similar label-DP guarantees when aggregating labels over random disjoint bags, while preserving the utility of Lipschitz-bounded neural mse-regression tasks.

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arXiv2026

Dense and Diverse Goal Coverage in Multi Goal Reinforcement Learning

Reinforcement Learning algorithms are primarily focused on learning a policy that maximizes expected return. As a result, the learned policy can exploit one or few reward sources. However, in many natural situations, it is desirable to learn a policy that induces a dispersed marginal state distribution over rewarding states, while maximizing the expected return which is typically tied to reaching a goal state. This aspect remains relatively unexplored. Existing techniques based on entropy regularization and intrinsic rewards use stochasticity for encouraging exploration to find an optimal policy which may not necessarily lead to dispersed marginal state distribution over rewarding states. Other RL algorithms which match a target distribution assume the latter to be available apriori. This may be infeasible in large scale systems where enumeration of all states is not possible and a state is determined to be a goal state only upon reaching it. We formalize the problem of maximizing the expected return while uniformly visiting the goal states as Multi Goal RL in which an oracle classifier over the state space determines the goal states. We propose a novel algorithm that learns a high-return policy mixture with marginal state distribution dispersed over the set of goal states. Our algorithm is based on optimizing a custom RL reward which is computed - based on the current policy mixture - at each iteration for a set of sampled trajectories. The latter are used via an offline RL algorithm to update the policy mixture. We prove performance guarantees for our algorithm, showing efficient convergence bounds for optimizing a natural objective which captures the expected return as well as the dispersion of the marginal state distribution over the goal states. We design and perform experiments on synthetic MDPs and standard RL environments to evaluate the effectiveness of our algorithm.

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arXiv2025

Algorithmic Guarantees for Distilling Supervised and Offline RL Datasets

Given a training dataset, the goal of dataset distillation is to derive a synthetic dataset such that models trained on the latter perform as well as those trained on the training dataset. In this work, we develop and analyze an efficient dataset distillation algorithm for supervised learning, specifically regression in $\mathbb{R}^d$, based on matching the losses on the training and synthetic datasets with respect to a fixed set of randomly sampled regressors without any model training. Our first key contribution is a novel performance guarantee proving that our algorithm needs only $\tilde{O}(d^2)$ sampled regressors to derive a synthetic dataset on which the MSE loss of any bounded linear model is nearly the same as its MSE loss on the given training data. In particular, the model optimized on the synthetic data has close to minimum loss on the training data, thus performing nearly as well as the model optimized on the latter. Complementing this, we also prove a matching lower bound of $Ω(d^2)$ for the number of sampled regressors showing the tightness of our analysis. Our second contribution is to extend our algorithm to offline RL dataset distillation by matching the Bellman loss, unlike previous works which used a behavioral cloning objective. This is the first such method which leverages both, the rewards and the next state information, available in offline RL datasets, without any policy model optimization. Our algorithm generates a synthetic dataset whose Bellman loss with respect to any linear action-value predictor is close to the latter's Bellman loss on the offline RL training dataset. Therefore, a policy associated with an action-value predictor optimized on the synthetic dataset performs nearly as well as that derived from the one optimized on the training data. We conduct experiments to validate our theoretical guarantees and observe performance gains.

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arXiv2025

Learning from Label Proportions and Covariate-shifted Instances

In many applications, especially due to lack of supervision or privacy concerns, the training data is grouped into bags of instances (feature-vectors) and for each bag we have only an aggregate label derived from the instance-labels in the bag. In learning from label proportions (LLP) the aggregate label is the average of the instance-labels in a bag, and a significant body of work has focused on training models in the LLP setting to predict instance-labels. In practice however, the training data may have fully supervised albeit covariate-shifted source data, along with the usual target data with bag-labels, and we wish to train a good instance-level predictor on the target domain. We call this the covariate-shifted hybrid LLP problem. Fully supervised covariate shifted data often has useful training signals and the goal is to leverage them for better predictive performance in the hybrid LLP setting. To achieve this, we develop methods for hybrid LLP which naturally incorporate the target bag-labels along with the source instance-labels, in the domain adaptation framework. Apart from proving theoretical guarantees bounding the target generalization error, we also conduct experiments on several publicly available datasets showing that our methods outperform LLP and domain adaptation baselines as well techniques from previous related work.

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arXiv2024

Aggregating Data for Optimal and Private Learning

Multiple Instance Regression (MIR) and Learning from Label Proportions (LLP) are learning frameworks arising in many applications, where the training data is partitioned into disjoint sets or bags, and only an aggregate label i.e., bag-label for each bag is available to the learner. In the case of MIR, the bag-label is the label of an undisclosed instance from the bag, while in LLP, the bag-label is the mean of the bag's labels. In this paper, we study for various loss functions in MIR and LLP, what is the optimal way to partition the dataset into bags such that the utility for downstream tasks like linear regression is maximized. We theoretically provide utility guarantees, and show that in each case, the optimal bagging strategy (approximately) reduces to finding an optimal clustering of the feature vectors or the labels with respect to natural objectives such as $k$-means. We also show that our bagging mechanisms can be made label-differentially private, incurring an additional utility error. We then generalize our results to the setting of Generalized Linear Models (GLMs). Finally, we experimentally validate our theoretical results.

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arXiv2024

Weak to Strong Learning from Aggregate Labels

In learning from aggregate labels, the training data consists of sets or "bags" of feature-vectors (instances) along with an aggregate label for each bag derived from the (usually {0,1}-valued) labels of its instances. In learning from label proportions (LLP), the aggregate label is the average of the bag's instance labels, whereas in multiple instance learning (MIL) it is the OR. The goal is to train an instance-level predictor, typically achieved by fitting a model on the training data, in particular one that maximizes the accuracy which is the fraction of satisfied bags i.e., those on which the predicted labels are consistent with the aggregate label. A weak learner has at a constant accuracy < 1 on the training bags, while a strong learner's accuracy can be arbitrarily close to 1. We study the problem of using a weak learner on such training bags with aggregate labels to obtain a strong learner, analogous to supervised learning for which boosting algorithms are known. Our first result shows the impossibility of boosting in LLP using weak classifiers of any accuracy < 1 by constructing a collection of bags for which such weak learners (for any weight assignment) exist, while not admitting any strong learner. A variant of this construction also rules out boosting in MIL for a non-trivial range of weak learner accuracy. In the LLP setting however, we show that a weak learner (with small accuracy) on large enough bags can in fact be used to obtain a strong learner for small bags, in polynomial time. We also provide more efficient, sampling based variant of our procedure with probabilistic guarantees which are empirically validated on three real and two synthetic datasets. Our work is the first to theoretically study weak to strong learning from aggregate labels, with an algorithm to achieve the same for LLP, while proving the impossibility of boosting for both LLP and MIL.

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arXiv2024

FRACTAL: Fine-Grained Scoring from Aggregate Text Labels

Large language models (LLMs) are being increasingly tuned to power complex generation tasks such as writing, fact-seeking, querying and reasoning. Traditionally, human or model feedback for evaluating and further tuning LLM performance has been provided at the response level, enabling faster and more cost-effective assessments. However, recent works (Amplayo et al. [2022], Wu et al. [2023]) indicate that sentence-level labels may provide more accurate and interpretable feedback for LLM optimization. In this work, we introduce methods to disaggregate response-level labels into sentence-level (pseudo-)labels. Our approach leverages multiple instance learning (MIL) and learning from label proportions (LLP) techniques in conjunction with prior information (e.g., document-sentence cosine similarity) to train a specialized model for sentence-level scoring. We also employ techniques which use model predictions to pseudo-label the train-set at the sentence-level for model training to further improve performance. We conduct extensive evaluations of our methods across six datasets and four tasks: retrieval, question answering, summarization, and math reasoning. Our results demonstrate improved performance compared to multiple baselines across most of these tasks. Our work is the first to develop response-level feedback to sentence-level scoring techniques, leveraging sentence-level prior information, along with comprehensive evaluations on multiple tasks as well as end-to-end finetuning evaluation showing performance comparable to a model trained on fine-grained human annotated labels.

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arXiv2024

Hardness of Learning Boolean Functions from Label Proportions

In recent years the framework of learning from label proportions (LLP) has been gaining importance in machine learning. In this setting, the training examples are aggregated into subsets or bags and only the average label per bag is available for learning an example-level predictor. This generalizes traditional PAC learning which is the special case of unit-sized bags. The computational learning aspects of LLP were studied in recent works (Saket, NeurIPS'21; Saket, NeurIPS'22) which showed algorithms and hardness for learning halfspaces in the LLP setting. In this work we focus on the intractability of LLP learning Boolean functions. Our first result shows that given a collection of bags of size at most $2$ which are consistent with an OR function, it is NP-hard to find a CNF of constantly many clauses which satisfies any constant-fraction of the bags. This is in contrast with the work of (Saket, NeurIPS'21) which gave a $(2/5)$-approximation for learning ORs using a halfspace. Thus, our result provides a separation between constant clause CNFs and halfspaces as hypotheses for LLP learning ORs. Next, we prove the hardness of satisfying more than $1/2 + o(1)$ fraction of such bags using a $t$-DNF (i.e. DNF where each term has $\leq t$ literals) for any constant $t$. In usual PAC learning such a hardness was known (Khot-Saket, FOCS'08) only for learning noisy ORs. We also study the learnability of parities and show that it is NP-hard to satisfy more than $(q/2^{q-1} + o(1))$-fraction of $q$-sized bags which are consistent with a parity using a parity, while a random parity based algorithm achieves a $(1/2^{q-2})$-approximation.

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arXiv2024

LLP-Bench: A Large Scale Tabular Benchmark for Learning from Label Proportions

In the task of Learning from Label Proportions (LLP), a model is trained on groups (a.k.a bags) of instances and their corresponding label proportions to predict labels for individual instances. LLP has been applied pre-dominantly on two types of datasets - image and tabular. In image LLP, bags of fixed size are created by randomly sampling instances from an underlying dataset. Bags created via this methodology are called random bags. Experimentation on Image LLP has been mostly on random bags on CIFAR-* and MNIST datasets. Despite being a very crucial task in privacy sensitive applications, tabular LLP does not yet have a open, large scale LLP benchmark. One of the unique properties of tabular LLP is the ability to create feature bags where all the instances in a bag have the same value for a given feature. It has been shown in prior research that feature bags are very common in practical, real world applications [Chen et. al '23, Saket et. al. '22]. In this paper, we address the lack of a open, large scale tabular benchmark. First we propose LLP-Bench, a suite of 70 LLP datasets (62 feature bag and 8 random bag datasets) created from the Criteo CTR prediction and the Criteo Sponsored Search Conversion Logs datasets, the former a classification and the latter a regression dataset. These LLP datasets represent diverse ways in which bags can be constructed from underlying tabular data. To the best of our knowledge, LLP-Bench is the first large scale tabular LLP benchmark with an extensive diversity in constituent datasets. Second, we propose four metrics that characterize and quantify the hardness of a LLP dataset. Using these four metrics we present deep analysis of the 62 feature bag datasets in LLP-Bench. Finally we present the performance of 9 SOTA and popular tabular LLP techniques on all the 62 datasets.

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arXiv2023

PAC Learning Linear Thresholds from Label Proportions

Learning from label proportions (LLP) is a generalization of supervised learning in which the training data is available as sets or bags of feature-vectors (instances) along with the average instance-label of each bag. The goal is to train a good instance classifier. While most previous works on LLP have focused on training models on such training data, computational learnability of LLP was only recently explored by [Saket'21, Saket'22] who showed worst case intractability of properly learning linear threshold functions (LTFs) from label proportions. However, their work did not rule out efficient algorithms for this problem on natural distributions. In this work we show that it is indeed possible to efficiently learn LTFs using LTFs when given access to random bags of some label proportion in which feature-vectors are, conditioned on their labels, independently sampled from a Gaussian distribution $N(\mathbfμ, \mathbfΣ)$. Our work shows that a certain matrix -- formed using covariances of the differences of feature-vectors sampled from the bags with and without replacement -- necessarily has its principal component, after a transformation, in the direction of the normal vector of the LTF. Our algorithm estimates the means and covariance matrices using subgaussian concentration bounds which we show can be applied to efficiently sample bags for approximating the normal direction. Using this in conjunction with novel generalization error bounds in the bag setting, we show that a low error hypothesis LTF can be identified. For some special cases of the $N(\mathbf{0}, \mathbf{I})$ distribution we provide a simpler mean estimation based algorithm. We include an experimental evaluation of our learning algorithms along with a comparison with those of [Saket'21, Saket'22] and random LTFs, demonstrating the effectiveness of our techniques.

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arXiv2022

Multi-Variate Time Series Forecasting on Variable Subsets

We formulate a new inference task in the domain of multivariate time series forecasting (MTSF), called Variable Subset Forecast (VSF), where only a small subset of the variables is available during inference. Variables are absent during inference because of long-term data loss (eg. sensor failures) or high -> low-resource domain shift between train / test. To the best of our knowledge, robustness of MTSF models in presence of such failures, has not been studied in the literature. Through extensive evaluation, we first show that the performance of state of the art methods degrade significantly in the VSF setting. We propose a non-parametric, wrapper technique that can be applied on top any existing forecast models. Through systematic experiments across 4 datasets and 5 forecast models, we show that our technique is able to recover close to 95\% performance of the models even when only 15\% of the original variables are present.

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arXiv2022

Tight Hardness of the Non-commutative Grothendieck Problem

$\newcommand{\eps}{\varepsilon} $We prove that for any $\eps > 0$ it is $\textsf{NP}$-hard to approximate the non-commutative Grothendieck problem to within a factor $1/2 + \eps$, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of $\ell_2$ into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight $\textsf{NP}$-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009).

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arXiv2019

Hardness of Learning DNFs using Halfspaces

The problem of learning $t$-term DNF formulas (for $t = O(1)$) has been studied extensively in the PAC model since its introduction by Valiant (STOC 1984). A $t$-term DNF can be efficiently learnt using a $t$-term DNF only if $t = 1$ i.e., when it is an AND, while even weakly learning a $2$-term DNF using a constant term DNF was shown to be NP-hard by Khot and Saket (FOCS 2008). On the other hand, Feldman et al. (FOCS 2009) showed the hardness of weakly learning a noisy AND using a halfspace -- the latter being a generalization of an AND, while Khot and Saket (STOC 2008) showed that an intersection of two halfspaces is hard to weakly learn using any function of constantly many halfspaces. The question of whether a $2$-term DNF is efficiently learnable using $2$ or constantly many halfspaces remained open. In this work we answer this question in the negative by showing the hardness of weakly learning a $2$-term DNF as well as a noisy AND using any function of a constant number of halfspaces. In particular we prove the following. For any constants $ν, ζ> 0$ and $\ell \in \mathbb{N}$, given a distribution over point-value pairs $\{0,1\}^n \times \{0,1\}$, it is NP-hard to decide whether, YES Case: There is a $2$-term DNF that classifies all the points of the distribution, and an AND that classifies at least $1-ζ$ fraction of the points correctly. NO Case: Any boolean function depending on at most $\ell$ halfspaces classifies at most $1/2 + ν$ fraction of the points of the distribution correctly. Our result generalizes and strengthens the previous best results mentioned above on the hardness of learning a $2$-term DNF, learning an intersection of two halfspaces, and learning a noisy AND.

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arXiv2017

Hardness of learning noisy halfspaces using polynomial thresholds

We prove the hardness of weakly learning halfspaces in the presence of adversarial noise using polynomial threshold functions (PTFs). In particular, we prove that for any constants $d \in \mathbb{Z}^+$ and $\varepsilon > 0$, it is NP-hard to decide: given a set of $\{-1,1\}$-labeled points in $\mathbb{R}^n$ whether (YES Case) there exists a halfspace that classifies $(1-\varepsilon)$-fraction of the points correctly, or (NO Case) any degree-$d$ PTF classifies at most $(1/2 + \varepsilon)$-fraction of the points correctly. This strengthens to all constant degrees the previous NP-hardness of learning using degree-$2$ PTFs shown by Diakonikolas et al. (2011). The latter result had remained the only progress over the works of Feldman et al. (2006) and Guruswami et al. (2006) ruling out weakly proper learning adversarially noisy halfspaces.

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arXiv2016

Approximation Algorithms for Stochastic k-TSP

We consider the stochastic $k$-TSP problem where rewards at vertices are random and the objective is to minimize the expected length of a tour that collects reward $k$. We present an adaptive $O(\log k)$-approximation algorithm, and a non-adaptive $O(\log^2k)$-approximation algorithm. We also show that the adaptivity gap of this problem is between $e$ and $O(\log^2k)$.

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arXiv2015

On the hardness of learning sparse parities

This work investigates the hardness of computing sparse solutions to systems of linear equations over F_2. Consider the k-EvenSet problem: given a homogeneous system of linear equations over F_2 on n variables, decide if there exists a nonzero solution of Hamming weight at most k (i.e. a k-sparse solution). While there is a simple O(n^{k/2})-time algorithm for it, establishing fixed parameter intractability for k-EvenSet has been a notorious open problem. Towards this goal, we show that unless k-Clique can be solved in n^{o(k)} time, k-EvenSet has no poly(n)2^{o(sqrt{k})} time algorithm and no polynomial time algorithm when k = (log n)^{2+eta} for any eta > 0. Our work also shows that the non-homogeneous generalization of the problem -- which we call k-VectorSum -- is W[1]-hard on instances where the number of equations is O(k log n), improving on previous reductions which produced Omega(n) equations. We also show that for any constant eps > 0, given a system of O(exp(O(k))log n) linear equations, it is W[1]-hard to decide if there is a k-sparse linear form satisfying all the equations or if every function on at most k-variables (k-junta) satisfies at most (1/2 + eps)-fraction of the equations. In the setting of computational learning, this shows hardness of approximate non-proper learning of k-parities. In a similar vein, we use the hardness of k-EvenSet to show that that for any constant d, unless k-Clique can be solved in n^{o(k)} time there is no poly(m, n)2^{o(sqrt{k}) time algorithm to decide whether a given set of m points in F_2^n satisfies: (i) there exists a non-trivial k-sparse homogeneous linear form evaluating to 0 on all the points, or (ii) any non-trivial degree d polynomial P supported on at most k variables evaluates to zero on approx. Pr_{F_2^n}[P(z) = 0] fraction of the points i.e., P is fooled by the set of points.

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arXiv2015

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{Ω\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.

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arXiv2013

Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs

This work revisits the PCP Verifiers used in the works of Hastad [Has01], Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable 4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to prove the following improved hardness results: Assuming that NP\not\subseteq DTIME(N^{O(loglog N)}), There is no polynomial time algorithm that, given an n-vertex 2-colorable 4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for some constant c > 0. There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for some constant c > 0. For any fixed k >= 4, there is no polynomial time algorithm that finds a partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0. Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs is an exponential improvement over the previous results of Guruswami et al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log n)^{-c} beyond the random assignment threshold for Max-E3-SAT and Max-Ek-Set-Splitting is an exponential improvement over the previous bounds proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results avoid the use of a variable bias parameter used in previous works, which leads to the improved hardness thresholds in addition to simplifying the analysis substantially. Apart from standard techniques from Fourier Analysis, for the first mentioned result we use a mixing estimate of Markov Chains based on uniform reverse hypercontractivity over general product spaces from the work of Mossel et al.[MOS13].

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